MAT 265 Final Exam Review (Spring 2008)



MAT 270 Test 3 Review (Spring 2012)

Test on April 11 in PSA 21

Section 3.7 Implicit Derivative

Remember: Equation of the tangent line through the point [pic] having slope m is

[pic].

Find the derivative [pic]

[pic] 2. [pic] 3. [pic] 4. [pic]

4. Find the equation of the tangent line to [pic] at the point x = 0.

5. Find the equation of the tangent line to [pic] at the point (1, 1).

Section 3.8 Derivative of Logarithmic and Exponential Functions

Remember: [pic] [pic] [pic]

Note: [pic]is a constant

Practice problems

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. [pic]

Section 3.9 Derivative of Inverse Trigonometric Functions

Remember the following:

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

Practice problems:

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic], simplify the answer.

7. Given [pic], find [pic]

8. The function[pic], [pic]has an inverse function g(x). Explain why. And find [pic]

9. The function [pic] has an inverse function g(x). Explain why. Find[pic]. Find equation of the tangent line to y = g(x) at (7, 1).

10. Find [pic], the derivative of the inverse function of g if [pic].

Section 3.10 Related Rates

Practice problems:

A puddle is evaporating in such a way that its diameter is decreasing at

rate of 0.1 cm/min. At what rate is the area of the puddle decreasing

when the diameter is 8 cm.

2. The volume of a cube is increasing at a rate of 10 cm3/min. How fast is

the surface area increasing when the length of the edge is 30 cm.

3. Two cars start moving from the same point. One travels South at 60mi/h

and the other travels West at 25mi/h. At what rate is the distance between

the two cars increasing two hours later?

4. A car is traveling at 50 mph due south at a point 1/2 mile north of an intersection. A police car is traveling at 40 mph due west at a point 1/4 mile east of the same intersection. At that instant, the radar in the police car measurers the rate at which the distance between the cars is changing. What does the radar gun register?

5. A 10-foot ladder leans against the side of a building. If the top of the ladder begins to slide down the wall at the rate of 2 ft/s, how fast is the bottom of the ladder sliding away from the wall when the top of the ladder is 8 ft off the ground.

6. An oil tanker has an accident and oil pours at the rate of 150 gallons per minute. Suppose that the oil spreads onto the water in a circle at a thickness of [pic]. Given that [pic]gallons. Determine the rate at which the radius of the spill is increasing when the radius reaches 500 ft.

Solution: The shape is like a cylinder. We consider the volume equation

[pic], where [pic]

Also given that [pic]

Thus [pic]

7. Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

Solution: We are given [pic], where [pic].

Then we get [pic].

8. A particle moves along the curve[pic]. As it reaches the point (2, 3), the y-coordinate is increasing at a rate of 4 cm/s. How fast is the x-coordinate of the point changing at that instance?

Solution: Given [pic], [pic] Find [pic] .

We write [pic]. By differentiation we get[pic]. Using the given values we get [pic].

4.1-4.4 Application/Optimization

1. Find the open interval where is the function [pic] concave up, concave down?

2. Find the local extrema of the function [pic]

3. Find all critical numbers of [pic] and also maximum and minimum values if any

4. Graph the function [pic]showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing.

5. Find the open interval where the function [pic] is increasing or decreasing. Find also local maximum and local minimum if any.

6. A farmer has 2500 ft of fencing and wants to fence of a rectangular field that borders a straight river. He needs not to fence along the river. What are the dimensions of the field that has the largest area?

7. Find the radius of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 6 inches and height 10 inches.

8. An 8 feet tall fence runs parallel to the side of a house 3 feet away. What is the length of the shortest ladder that clears the fence and reaches the house? Assume that the vertical wall of the house and the horizontal ground have infinite extent. Answer: Length approximately 15 ft.

9. The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $800 per month. A market survey suggests that, on average, one additional unit will remain vacant for each $10 increase in rent. What rent should the manager charge to maximize revenue?

10. A right circular cylinder is inscribed in a sphere of radius r. To find the largest possible surface area of such a cylinder.

11. A 8000 cubic foot tank with a square base and an open top is to be constructed of a sheet of steel of a given thickness. Find the length of a side of the square base of the tank with minimum surface area.

12. Find the x-coordinate(s) for the points on the graph of [pic]that are closest to the point (0, 5).

13. Find the maximum area of the circle inscribed in an equilateral triangle whose sides have length 2 meters. Round your answer correct to two decimal places if needed.

Section 4.5 Linear Approximation

We have the linear approximation [pic]

Practice problems:

1. Find the linear approximation of [pic] at [pic] and thus approximate the values of [pic] and [pic]

2. Evaluate [pic] using Linear Approximation

3. Prove that[pic] for x close to 0, and illustrate this approximation by drawing the graphs of [pic] and [pic] on the same screen.

We consider [pic]

Now using linear approximation formula we find

[pic]

Section 4.6 Mean Value Theorem

First of all we discuss the special case of Mean Value Theorem.

The Rolle’s Theorem: Suppose that f is continuous on the interval [pic], differentiable on [pic]and [pic]then there is a number [pic]such that [pic]

The Mean Value Theorem: Suppose that f is continuous on the interval [pic], differentiable on [pic]then there is a number [pic]such that [pic]

Constant Theorem: Suppose that [pic]for all x on an open interval I, then f must be a constant, that is [pic], a constant

Corollary: Suppose that [pic]for all x in some open interval I, then for some constant c, [pic]

1. Determine whether Rolle’s Theorem applies to the function [pic]on [0, 10]. If so, find the points that are guaranteed to exist by Rolle’s Thereom.

2. Determine whether Rolle’s Theorem applies to the function [pic]on [0, 10]. If so, find the points that are guaranteed to exist by Rolle’s Thereom.

3. Determine whether Mean Value Theorem applies to the function [pic]on [0, 9].

Section 4.7 L’Hopitals Rule

We have noticed the following indeterminate forms [pic]. The L’Hopital rule is applicable for [pic] if this limit has the form [pic]. The L’Hopital rule is as follows [pic]. One may apply the rule repeatedly as long as the indeterminate form appears.

Examples with the above indeterminate forms:

1. [pic]

2. [pic], does not exist (DNE)

3. [pic]

4. [pic], DNE as the result is not a finite number

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic] DNE

Section 4.8 Integration

1. Find f(x) if [pic]

2. Find f(x) if [pic]

3. Find g(x) when [pic]

4. Find f(x), given [pic]

Find the following indefinite integral

[pic]

[pic]

[pic]

[pic]

[pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

Answer: Section 3.7: 1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. [pic]

Section 3.8: 1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. 2 11. [pic] 12. [pic]

13. [pic]

Section 3.9: 1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic] 7. 1/3

8. ½ 9. [pic] 10. 1/76

Section 3.10: 1. [pic] 2. [pic] 3. 65 mph

Section 4.1-4.4: 1. Up on [pic] 2. Max ( 0.8, 1.03) 3. Max (-2, 30.24); min(2, -30.24)

4. Min(1.6, -10.46), do the rest by yourself

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